本科毕业设计（论文）
外文翻译
不等式
作者：G.H.Hardy, J.E.Littlewood, G.Polya
国籍：英国
出处：Inequalities
中文译文：
1.1. 有限、无限和积分不等式
将一些特殊的或者典型的不等式作为本章的一般性研究对象会比较方便；在这里我们首先选择了Cauchy的一个重要定理，其通常被称为“Cauchy不等式”
Cauchy不等式（定理7）为
或者
(1.1.2)
柯西不等式对取所有实值都成立. 我们称为不等式的变量，这里变量的数目是有限的. 另外不等式也陈述了某些有限和之间的关系. 我们称这样的不等式为初等不等式或有限不等式.
尽管最基本的不等式的变量数目是有限的，但是我们也要关注非有限的不等式和涉及到求和概念的推广. 这些普遍原理中最重要的是无穷和
(1.1.3) ,
和积分
(1.1.4)
（其中和可以是有限的也可以是无限的）. 则对应到(1.1.2)，有相似定理
(1.1.5)
（或求和的上下极限都是无穷大时有类似的公式）和
(1.1.6)
我们称(1.1.5)为无限不等式，称(1.1.6)为积分不等式.
2.13．基本不等式的说明和应用
(i)霍尔德和闵可夫斯基不等式的几何解释
霍尔德不等式和闵可夫斯基不等式两个特别简单的例子分别是
(2.13.1)
(2.13.2)
其中对变量取所有实值都成立，且表述如下事实：(1)实角的余弦值在数值上小于1；(2)三角形两条边的和大于第三条边. 例外情况是：(1)向量和平行(方向相同或相反)，(2)向量平行且相等.
闵可夫斯基不等式的一般形式是将(2.13.2)推广到n维空间，也是距离的一般定义，即
(2.13.1)最直接的拓展不是关于针对一般r的Holder不等式，而与r=2时在不同方向上的一般化有关.
29.如果当时是正二次型(系数为实数，但不一定是正数)，则
如果(x)和(y)成比例则不成立.
这是的直接结果.
这是显然的：与定理7的第二个证明相比较. 它从几何方面在斜坐标或非欧几里得度量标准上表示了(2.13.1)在n维空间上的扩展.
为证明定理15，取和直角坐标. 根据这个定理可以断言，如果一个向量沿任意方向的投影长度都不超过l，那么这个向量的长度也不会超过l.
(ii)阿达玛定理
在我们的下一个定理中，我们还涉及到一组数字实数，它们是实数，但不一定是正数.
30.设D是一个行列式，它的元素是
则
(2.13.3)
当且仅当下式成立时取等号
(2.13.4)
适用于任一对不同的，或是当式子(2.13.3)右边的一个因子为零时成立.
该定理的几何意义是，在n维空间中，平行六面体的体积不会超过从任一角向各顶点发散出的线向量之积，且只有当这些线向量是正交的或者一条边不存在时式子(2.13.3)才会相等.
假设当时，是一个正二次型，是组成元素为的行列式，那么方程
(2.13.5)
有n个正根，且它们的和为，乘积为.
因此，根据定理9，可以得到
（2.13.6)
如果对于任意都有，则以下式子
也成立；如果我们把(2.13.6)代入到这个式子，我们得到
(2.13.7)
这个式子在本质上等价于阿达玛定理. 下列式子
也成立（除非D=0），且有
所以式子 (2.13.7) 即为 (2.13.3).
对于(2.13.6)中的等式成立时，式子(2.13.5)中所有的根一定是相等的，这种情况当且不依赖于时才有可能. 因此，对于(2.13.7)中的等式，我们得到且 不依赖于时有，因为有且由得所以最后一个条件肯定是满足的，即(2.13.4)中等式成立.
我们可以用Hermitian形式代替二次型，将这个定理推广到更复杂的行列式中. Schur(2)进行了进一步的扩展.
Oppenheim对(2.13.7)做了巧妙的证明. Oppenheim的论证不仅得到了式子(2.13.7)和Hadamard定理，还建立了下面的不等式(2.13.8)和(2.13.9)，这两个不等式分别是由Minkowski和Fischer提出的.
任何两个正二次型, ，都可以通过行列式的线性变换转化为平方和，如,，其中和 是正的. 那么被简化为，而行列式, ...符合的表格
,,
因此，将几何代入定理10，可以得到
(2.13.8)
假设d的矩阵是由c的矩阵相乘得到的，首先通过方式一对前r行进行变换，然后是对前r列进行变换. 接下来如果把式子(2.13.8)分成两份，取n次方，那么可以得到
(2.13.9)
在这个式子中，为中r行r列的顺序主子式（西北主子式），为东南余子式. 按照这种方式，用两个因数替换式子(2.13.9)右边的每个因数，以此类推，最终得到(2.13.7).
(iii)矩阵的模
假设A和B是都是n行n列的矩阵，其元素分别是和；这些元素可能很复杂. 定义矩阵A B和BA的元素分别为
,.
31.设 是矩阵A的模，则它可以定义为 ，
那么
,
第一个不等式令定理25中r=2可以直接得出. 第二个可以由定理7得出，因为
(iv)初等几何中的极大值和极小值
我们引用了一些基本不等式在初等几何上的许多应用问题(作为读者的练习).
32. 如果给定周长2p，则当边a、边b、边c相等时，三角形的面积是最大的.
[将p-a, p-b, p-c代入定理9]
33. 如果给定一个平行六面体的曲面，则该平行六面体为正方体时，其体积最大.
[表示由a, b, c从任一角发散出的边，可以将bc, ca, ab代入定理9. 对于n维的平行六面体，有一个类似的定理；如果klt;n并且k维边界的曲面已知，当平行六面体为矩形且其边长相等时，其体积最大. 这可以通过将定理9、定理30与行列式之间的恒等式来结合证明.
附：外文原文
fundamental inequalities
1.1. Finite, infinite, and integral inequalities.
It will be convenient to take some particular and typical inequality as a text for the general remarks which occupy this chapter; and we select a remarkable theorem due to Cauchy and usually known as rsquo;Cauchyrsquo;s inequalityrsquo;.
Cauchyrsquo;s inequality (Theorem 7) is
or
(1.1.2)
and is true for all real values of. We call the variables of the inequality. Here the number of variables is finite, and the inequality states a relation between certain finite sums. We call such an inequality an elementary or finite inequality.
The most fundamental inequalities are finite, but we shall also be concerned with inequalities which are not finite and involve generalisations of the notion of a sum. The most important of such generalisations are the infinite sums
(1.1.3) ,
and the integral
(1.1.4)
(where and may be finite or infinite). The analogues of (1.1.2) corresponding to these generalisations are
(1.1.5)
(or the similar formula in which both limits of summation are infinite), and
(1.1.6)
We call (1.1.5) an infinite, and (1.1.6) an integral, inequality.
2.13. Illustrations and applications of the fundamental inequalities.
(i)Geometrical interpretations of Holderrsquo;s and Minkowskirsquo;s inequalities.
Two particularly simple cases of Holderrsquo;s and Minkowskirsquo;s inequalities are
(2.13.1)
(2.13.2)
These hold for all real values of the variables, and express the facts that (1) the cosine of a real angle is numerically less than 1, And (2) the sum of two sides of a triangle is greater than the third side. The exceptional cases are those in which (1) the vectorsandare parallel (with the same or opposite senses), and (2) the vectors are parallel and have the same sense.
The ordinary form of Minkowskirsquo;s inequality is the extension of (2.13.2) to space
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fundamental inequalities
1.1. Finite, infinite, and integral inequalities.
It will be convenient to take some particular and typical inequality as a text for the general remarks which occupy this chapter; and we select a remarkable theorem due to Cauchy and usually known as rsquo;Cauchyrsquo;s inequalityrsquo;.
Cauchyrsquo;s inequality (Theorem 7) is
or
(1.1.2)
and is true for all real values of. We call the variables of the inequality. Here the number of variables is finite, and the inequality states a relation between certain finite sums. We call such an inequality an elementary or finite inequality.
The most fundamental inequalities are finite, but we shall also be concerned with inequalities which are not finite and involve generalisations of the notion of a sum. The most important of such generalisations are the infinite sums
(1.1.3) ,
and the integral
(1.1.4)
(where and may be finite or infinite). The analogues of (1.1.2) corresponding to these generalisations are
(1.1.5)
(or the similar formula in which both limits of summation are infinite), and
(1.1.6)
We call (1.1.5) an infinite, and (1.1.6) an integral, inequality.
2.13. Illustrations and applications of the fundamental inequalities.
(i)Geometrical interpretations of Holderrsquo;s and Minkowskirsquo;s inequalities.
Two particularly simple cases of Holderrsquo;s and Minkowskirsquo;s inequalities are
(2.13.1)
(2.13.2)
These hold for all real values of the variables, and express the facts that (1) the cosine of a real angle is numerically less than 1, And (2) the sum of two sides of a triangle is greater than the third side. The exceptional cases are those in which (1) the vectorsandare parallel (with the same or opposite senses), and (2) the vectors are parallel and have the same sense.
The ordinary form of Minkowskirsquo;s inequality is the extension of (2.13.2) to space of n dimensions with a generalised definition of distance, viz.
The most obvious extensions of (2.13.1) are connected not with Holderrsquo;s inequality for general r but with a generalisation of the case r=2 in a different direction.
29. If where is a positive quadratic form（with real, but not necessarily positive, coefficients), then
unless (x) and (y) are proportional.
This is an immediate consequence of the fact that
is positive: compare the second proof of Theorem 7. It represents geometrically an extension of (2.13.1) to n-dimensional space, with oblique coordinates or a non-Euclidean metric.
To illustrate Theorem 15, take and rectangular coordinates. The theorem then asserts that, if the length of the projection of a vector along an arbitrary direction does not exceed , the length of the vector does not exceed .
(ii)A theorem of Hadamard.
In our next theorem also we are concerned with a set of numbers real but not necessarily positive.
30. If D is the determinant whose constituents are
then
(2.13.3)
There is equality only when
(2.13.4)
for every distinct pair,or when one of the factors on the righthand side of (2.13.3) vanishes.
The geometrical significance of the theorem is that the volume of a parallelepiped in n-space does not exceed the product of the edges diverging from one corner, and that there is equality only when they are orthogonal or an edge vanishes.
Suppose that where , is a positive quadratic form, and that is the determinant whose constituents are , Then the equation
(2.13.5)
has n positive roots whose sum is and whose product is .
Hence, by Theorem 9,
(2.13.6)
If for all , then the form
is also positive; and if we apply (2.13.6) to this form, we obtain
(2.13.7)
This is substantially equivalent to Hadamardrsquo;s theorem. For the form
is positive unless D=0. Also and
so that (2.13.7) is (2.13.3).
For equality in (2.13.6) , all the roots of (2.13.5) must be equal, which is only possible if whenever and is independent of. Hence, for equality in (2.13.7) , we must have for , independent of . The last condition is certainly satisfied, since, and is , which is (2.13.4).
We can extend the theorem to determinants with complex constituents by using Hermitian instead of quadratic forms. Further extensions have been made by Schur (2).
The following ingenious proof of (2.13.7) is due to Oppenheim. Oppenheimrsquo;s argument establishes not only (2.13.7), and so Hadamardrsquo;s theorem, but also the inequalities (2.13.8) and (2.13.9) below, due to Minkowski and Fischer respectively.
Any two positive quadratic forms,may be reduced simultaneously, by a linear transformation of determinant unity, to sums of squares, say,where and are positive. Then is reduced to and the determinants, ... of the forms satisfy
,,
Hence, applying Theorem 10 to the sets, we obtain
(2.13.8)
Suppose now that the matrix of the d is formed from that of the c by multiplying, first the first r rows, and then the first r columns, by-1. If then we divide (2.13.8) by 2, and raise to the nth power, we obtain
(2.13.9)
Where denotes the north-west diagonal minor of r rows and columns in , anddenotes
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